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Your data matches 48 different statistics following compositions of up to 3 maps.
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Matching statistic: St000325
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [1,2] => [1,2] => 1
[1,1,0,0]
=> [2,1] => [2,1] => 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => 2
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 3
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 3
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => 4
Description
The width of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]].
See also [[St000308]] for the height of this tree.
Matching statistic: St000470
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [1,2] => [1,2] => 1
[1,1,0,0]
=> [2,1] => [2,1] => 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => 2
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 3
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 3
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => 4
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000021
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => 2 = 3 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => 3 = 4 - 1
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000007
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0]
=> [1,2] => [2,1] => [2,1] => 2
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => [3,2,1] => 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => [2,3,1] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => [3,1,2] => 2
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => [1,3,2] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => [1,3,2] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => [3,4,2,1] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 3
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => [2,4,3,1] => 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,3,4,1] => [2,4,3,1] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,3,4,2] => [1,4,3,2] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,1,2,3] => [4,1,3,2] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,4,2,3] => [1,4,3,2] => 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,3,2,4] => [1,4,3,2] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => [1,4,3,2] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => [4,5,3,2,1] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => [3,5,4,2,1] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,4,5,2,1] => [3,5,4,2,1] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => [4,5,2,3,1] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => [2,5,4,3,1] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [2,4,5,3,1] => [2,5,4,3,1] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [5,2,3,4,1] => [5,2,4,3,1] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,5,3,4,1] => [2,5,4,3,1] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [2,4,3,5,1] => [2,5,4,3,1] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,3,4,5,1] => [2,5,4,3,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,1,2] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [4,5,3,1,2] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => [5,3,4,1,2] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => [3,5,4,1,2] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,4,5,1,2] => [3,5,4,1,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => [5,4,1,3,2] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => [4,5,1,3,2] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => [5,1,4,3,2] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,4,5,3,2] => [1,5,4,3,2] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [5,1,3,4,2] => [5,1,4,3,2] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,5,3,4,2] => [1,5,4,3,2] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,4,3,5,2] => [1,5,4,3,2] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,3,4,5,2] => [1,5,4,3,2] => 4
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000015
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00005: Alternating sign matrices —transpose⟶ Alternating sign matrices
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00005: Alternating sign matrices —transpose⟶ Alternating sign matrices
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1]]
=> [[1]]
=> [1,0]
=> 1
[1,0,1,0]
=> [[1,0],[0,1]]
=> [[1,0],[0,1]]
=> [1,0,1,0]
=> 2
[1,1,0,0]
=> [[0,1],[1,0]]
=> [[0,1],[1,0]]
=> [1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,0,1,1,0,1,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
Description
The number of peaks of a Dyck path.
Matching statistic: St000393
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1 => 1 => 1
[1,0,1,0]
=> [1,1] => 11 => 11 => 2
[1,1,0,0]
=> [2] => 10 => 01 => 1
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 111 => 3
[1,0,1,1,0,0]
=> [1,2] => 110 => 011 => 2
[1,1,0,0,1,0]
=> [2,1] => 101 => 101 => 2
[1,1,0,1,0,0]
=> [3] => 100 => 001 => 2
[1,1,1,0,0,0]
=> [3] => 100 => 001 => 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 1111 => 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 0111 => 3
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 1011 => 3
[1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 0011 => 3
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 0011 => 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 1101 => 3
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 0101 => 2
[1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 1001 => 3
[1,1,0,1,0,1,0,0]
=> [4] => 1000 => 0001 => 3
[1,1,0,1,1,0,0,0]
=> [4] => 1000 => 0001 => 3
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1001 => 3
[1,1,1,0,0,1,0,0]
=> [4] => 1000 => 0001 => 3
[1,1,1,0,1,0,0,0]
=> [4] => 1000 => 0001 => 3
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 0001 => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 11111 => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 01111 => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 10111 => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 11100 => 00111 => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 00111 => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 11011 => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 01011 => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 11001 => 10011 => 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 00011 => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 00011 => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 10011 => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 00011 => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 00011 => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 00011 => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 11101 => 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 01101 => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 10101 => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 00101 => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 00101 => 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 11001 => 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 01001 => 3
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 10001 => 4
[1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 00001 => 4
[1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 00001 => 4
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 10001 => 4
[1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 00001 => 4
[1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 00001 => 4
[1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 00001 => 4
Description
The number of strictly increasing runs in a binary word.
Matching statistic: St000553
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000553: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000553: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 1
[1,0,1,0]
=> [2,1] => ([],2)
=> ([],2)
=> 2
[1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => ([],3)
=> ([],3)
=> 3
[1,0,1,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([],4)
=> ([],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 3
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([],5)
=> ([],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(3,4)],5)
=> ([(3,4)],5)
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
Description
The number of blocks of a graph.
A cut vertex is a vertex whose deletion increases the number of connected components. A block is a maximal connected subgraph which itself has no cut vertices. Two distinct blocks cannot overlap in more than a single cut vertex.
Matching statistic: St001068
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00005: Alternating sign matrices —transpose⟶ Alternating sign matrices
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00005: Alternating sign matrices —transpose⟶ Alternating sign matrices
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1]]
=> [[1]]
=> [1,0]
=> 1
[1,0,1,0]
=> [[1,0],[0,1]]
=> [[1,0],[0,1]]
=> [1,0,1,0]
=> 2
[1,1,0,0]
=> [[0,1],[1,0]]
=> [[0,1],[1,0]]
=> [1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,0,1,1,0,1,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St000053
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00005: Alternating sign matrices —transpose⟶ Alternating sign matrices
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00005: Alternating sign matrices —transpose⟶ Alternating sign matrices
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1]]
=> [[1]]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [[1,0],[0,1]]
=> [[1,0],[0,1]]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [[0,1],[1,0]]
=> [[0,1],[1,0]]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
Description
The number of valleys of the Dyck path.
Matching statistic: St000155
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [3,2,1] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [2,3,1] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => [1,3,4,2] => 2 = 3 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => [1,3,4,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => [1,3,4,2] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => [4,2,1,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => [3,2,4,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => [3,2,4,1] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [4,3,2,1] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [2,4,3,1] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [2,3,4,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [1,3,4,5,2] => 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [1,3,4,5,2] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [1,3,4,5,2] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [1,3,4,5,2] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => [1,3,4,5,2] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [1,3,4,5,2] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [1,3,4,5,2] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [1,3,4,5,2] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [1,3,4,5,2] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => [1,3,4,5,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => [1,3,4,5,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => [1,3,4,5,2] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => [1,3,4,5,2] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,3,4,5,2] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => [2,1,4,5,3] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => [2,1,4,5,3] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => [2,1,4,5,3] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => [5,2,4,1,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => [5,2,4,1,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => [4,2,1,5,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => [3,2,4,5,1] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => [3,2,4,5,1] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => [4,2,1,5,3] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => [3,2,4,5,1] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => [3,2,4,5,1] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => [3,2,4,5,1] => 3 = 4 - 1
Description
The number of exceedances (also excedences) of a permutation.
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
The following 38 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000157The number of descents of a standard tableau. St000245The number of ascents of a permutation. St000632The jump number of the poset. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001298The number of repeated entries in the Lehmer code of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St000288The number of ones in a binary word. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001875The number of simple modules with projective dimension at most 1. St001427The number of descents of a signed permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000619The number of cyclic descents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001668The number of points of the poset minus the width of the poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000264The girth of a graph, which is not a tree. St000454The largest eigenvalue of a graph if it is integral. St001626The number of maximal proper sublattices of a lattice. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000862The number of parts of the shifted shape of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one.
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